Physics Letters B 666 (2008) 193–198

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Physics Letters B

www.elsevier.com/locate/physletb

Inclusion of Yang–Mills fields in string corrected supergravity

S. Bellucci a,∗, D. O’Reilly b

a INFN-Laboratori Nazionali di Frascati, Via E. Fermi 40, 00044 Frascati, Italyb Department of Mathematics, The Graduate School and University Center, 365 Fifth Avenue, New York, NY 10016-4309, USA

a r t i c l e i n f o a b s t r a c t

Article history:Received 27 May 2008Received in revised form 30 June 2008Accepted 3 July 2008Available online 5 July 2008Editor: T. Yanagida

PACS:04.65.+e

We consistently incorporate Yang–Mills matter fields into string corrected (deformed) D = 10, N = 1supergravity. We solve the Bianchi identities within the framework of the modified beta function favoredconstraints to second order in the string slope parameter γ also including the Yang–Mills fields. In thetorsion, curvature and H sectors we find that a consistent solution is readily obtained with a Yang–Mills modified supercurrent Aabc . We find a solution in the F sector following our previously developedmethod.

© 2008 Elsevier B.V. All rights reserved.

1. Introduction

String corrections to quantum field theories are believed to contribute Gauss–Bonnet terms to the action. This invariant has beenstudied in many different scenarios, for example Gauss–Bonnet modified cosmology [1]. These terms are introduced by hand into thequantum gravity models. However it is also known that Gauss–Bonnet terms occur naturally in the context of string corrected gravity [2].The leading order corrections in terms of the string tension parameter, γ contain these invariants. Hence we are motivated by this and byother reasons to study string deformed supergravity, with D = 10, N = 1 as the low energy limit of string theory. Here we wish to includeYang–Mills fields in the non-minimal theory. As several of the terms are extremely lengthy we avoid writing them explicitly. In a futurework we will explore the simplified bosonic case.

Some years ago a scenario was developed to construct a manifestly supersymmetric theory of string corrected supergravity [2–6].This involved incorporating the Lorentz Chern–Simons superform, X LL into the geometry of D = 10, N = 1 supergravity. It was initiallysuccessful at first order in γ [2]. It ran into difficulties and controversy at second order [7]. It was suggested that at second order thetorsion Tαβ

g should be modified to include the so-called X tensor [3]. A search for the X tensor ensued and a candidate was proposed in[8] which was shown to allow for the solution of the Bianchi identities in the H sector and also the torsion and curvature sectors.

In this Letter we show that a simple modification of the X tensor Ansatz allows also for the inclusion of matter fields in the H torsionand curvature sectors. We show that the assumption F (2)

αβ = 0 does not allow for a solution. We propose a candidate that does allow fora solution. Prior to finding the X tensor also by way of an Ansatz, in [8], the search for a second order solution consisted of systematicallystudying the table of irreducible representations [9]. However the task proved formidable because of the number of unknown quantities.It was eventually shown that the form given in Eq. (33) of the first reference of [8] for T (2)

αβg , satisfying the torsion identity at dimension

one half, thereafter allowed ultimately for a consistent solution in the torsion and curvature sectors.In Ref. [8], we found a solution to D = 10, N = 1 supergravity to second order in the string slope parameter with the modified tensor

G ADG = H ADG + γ Q ADG . (1)

Here we extend this as follows:

G ADG = H ADG + γ Q ADG + βY ADG . (2)

Here β is the Yang–Mills coupling constant. Matter fields will not have consequences for the appearance of the Gauss–Bonnet termhowever, but we wish to construct a complete model.

* Corresponding author.E-mail addresses: bellucci@lnf.infn.it (S. Bellucci), doreilly@gc.cuny.edu, dunboyne@vzavenue.net (D. O’Reilly).

0370-2693/$ – see front matter © 2008 Elsevier B.V. All rights reserved.doi:10.1016/j.physletb.2008.07.004

194 S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198

2. The solution

The Bianchi identities in superspace are as follows:[[∇[A,∇B},∇C)

} = 0. (3)

Here we have extended the commutator in [8] to include the Yang–Mills field strengths F IAB

[∇A,∇B} = T ABC ∇C + 1

2R AB

de Med + i F IABtI . (4)

The t I are the generators of the Yang–Mills gauge group. For notation convenience we drop the group index I . For notational brevity wealso write

R ABde = R(0)ABde + R(1)

ABde + R(2)ABde + · · · (5)

and

T ADG = T (0)

ADG + T (1)

ADG + T (2)

ADG + · · · . (6)

Where the numerical superscript refers to the order of the quantity. A quantity which re-occurs is the following

Ω(1)gef = L(1)

gef − 1

4A(1)

gef (7)

and its spinor derivative which we denote simply as

Ω(1)γ gef = ∇γ

[L(1)

gef − 1

4A(1)

gef

]. (8)

We evaluate this in Appendix A. Ref. [2] began with the conventional constraints as follows:

iσbαβ Tαβ

a = 16δab, σabcde

αβ Tαβa = 0, σa

αβ Tαβa = 0,

Tα[bc] = 0, Tabc = 1

6T [abc], Taα

β = 1

48σa αλσ

pqrβλ A pqr . (9)

Using these constraints then led to the first order solution. That is G(1)ABC , T (1)

ABC , and R(1)

ABde were found, along with additionalmodified constraints. In [8] we found that no modification to the condition Hαβγ = 0 was necessary. We continue with this assumption.

We have the H sector Bianchi identities with Yang–Mills fields as follows [2]:

1

6∇(α|H |βγ δ) − 1

4T(αβ|M HM|γ δ) = −γ

4R(αβ|ef R |γ δ)

ef − β

4F(αβ| I F |γ δ)

I , (10)

1

2∇(α|H |βγ )d − ∇d Hαβγ − 1

2T(αβ|M HM|γ )d + 1

2Td(α|M HM|βγ ) = −γ R(αβ|ef R |γ )d

ef − β F(αβ| I F |γ )dI , (11)

∇(α|Hβ)cd + ∇[c|H |d]αβ − TαβM HMcd − Tcd

M HMαβ − T(α|[c|M HM|d]|β)

= −γ[2Rαβef Rcd

ef + R(α|[c|ef R |d]β)ef ] − β

[2Fαβ

I FcdI + F(α|[c| I F |d]β)

I]. (12)

We must also solve the torsions and curvatures

T(αβ|λT |γ )λd − T(αβ| g T |γ )g

d − ∇(α|Tβγ )d = 0, (13)

T(αβ|λT |γ )λδ − T(αβ| g T |γ )g

δ − ∇(α|T |βγ )δ − 1

4R(2)

(αβ|deσde |γ )

δ = 0, (14)

T(αβ|λR |γ )λde − T(αβ|λR |γ )λde − ∇(α|R |βγ )de = 0. (15)

To this list we now add the equations arising form the F sector

∇[A| F |BC) − T M [AB| F M|C) = 0. (16)

In our previous work the supercurrent A(1)gef was given by

A(1)gef = +iγ σgef ετ T mnε Tmn

τ . (17)

To begin with we modify A(1)gef to include matter fields as in [2].

A(1)gef −→ +iσgef ετ

[γ T mnε Tmn

τ + βλελτ] = A(γ )

gef + A(β)gef . (18)

Here we use the notation where the superscript γ or β is self evident. This has the effect of splitting the Bianchi identities howeverwe still have to be careful of cross terms. In order to be cautious we will examine all contributions in the H sector, in particular thatfor H(2)

aβd , for thoroughness (see Appendix A). In all of the Bianchi identities we encounter the spinor derivative ∇α A(1)abc . We now

show that a key equation which led to the previous second order modified beta function favored (β F F ) solution is still valid but with themodified A(1)

abc . The spinor derivative of A(1)abc will now contain the contributions due to the spinor derivative of λτ at zeroth order.

We have the following modifications from [2]:

S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198 195

∇γ T (0)ef

δ = −1

4σmn

γδ R(0)

ef mn + T (0)ef

λT (0)γ λ

δ, (19)

∇γ λ(0)δ = −1

4σmn

γδ Fmn + λλT (0)

γ λδ. (20)

The fundamental equation which enable a solution to be found with higher order β F F constraints, now with the modified supercurrent,A(1)

pqr is still given by

T (0)(αβ|λσ pqref |γ )λ A(1)

pqr H(0)def − σ pqref

(αβ|H(0)def ∇|γ ) A(1)

pqr = −24σ g(αβ|H(0)

def [Ω(1)|γ )gef

]. (21)

In the H sector we will have the following quantities

H ABC = H(γ γ )ABC + H(ββ)

ABC + H(βγ )ABC . (22)

We will also adopt the modified torsion

T (2)αβ

g = − iγ

6σ pqref

αβ H(0)def A(1)

pqr . (23)

We then find no change in the form of the H sector results.

H̄(2)αβd = H(γ β)

αβd = σαβg[−iγ H(0)

def A(β)g

ef ] − σ pqrefαβ

[iγ

12H(0)

def A(β)pqr

]. (24)

Eq. (21) contains the second order contribution of the spinor derivative ∇γ Hαβg . Term by term we notice no order β2 contributionsoccur. We also seek terms of the form O (γ β) and note that none appear other than those resulting from the modification to Aabc . Wealso find H(2)

gγ d as in [8], but we write it in a different way

+ i

2σ(αβ|g H(2)

g|γ )d = − γ

12σγ |

(αβ|σ pqrg

f |γ )λ A(1)pqr Tdf

λ + iσ g(αβ|

{4γ

[∇|γ )H(0)d

ef ]L(1)gef + 4γ H(0)

def ∇|γ )L(1)

gef

− γ

2

[∇|γ )H(0)d

ef ]A(1)gef − γ

2H(0)

def ∇|γ ) A(1)

gef − 2γ H(0)g

ef R(1)|γ )def − 2γ L(1)g

ef R(0)|γ )def

+ γ

4A(1)

gef R(0)|γ )def + 2γ∇|γ )

[H(0)

def H(0)

gef]Order(1)

}. (25)

The symmetrized result can be extracted as in [2]. We list it in Appendix A. We also believe that it can be simplified further. We alsohave, along with (33) of the first reference of [8],

iσ g(αβ|T (2)|γ )g

δ = +2iγ σ g(αβ|T (0)ef δΩ(1)|γ )gef , (26)

+iσ g(αβ|T (2)|γ )gd = +4iγ σ g

(αβ|H(0)d

ef [Ω(1)|γ )gef] + γ

6σ g

(αβ|σ pqreg|γ )φ A(1)

pqr T (0)de

φ, (27)

Rαβde = − iγ

12σ pqref

αβ A(1)pqr Ref de. (28)

3. F sector Bianchi identities

We have seen that in the H and torsion sectors the required minimal Bianchi identities have been fully satisfied by simply modifyingthe supercurrent as in Eq. (28). The fundamental identity (21) is then used to solve in each case. This is the identity that enables suchsolutions to be obtained in the modified β function favored set of constraints. Here we show that it can also be used to solve in the Fsector coupled with an Ansatz. To begin with we consider the following Bianchi identities:

T(αβ|λ F |γ )λ − T(αβ| g F |γ )g − ∇(α| F |βγ ) = 0, (29)

∇α Fab − ∇[a| Fα|b] − Tα[a|M F M|b] − TabM F M α = 0. (30)

We have from [2] to first order,

F (0)αβ = F (1)

αβ = F (1)αd = 0, (31)

F (0)αd = −iσdαφλφ, (32)

∇α F (0)ef = iσ[e|αφ∇| f ]λφ − 2iσ g

αφ H(0)gef . (33)

Eq. (29) is not satisfied by F (2)αβ = 0. We propose the following candidate and show that it works.

F (2)αβ = − iγ

12σ pqref

αβ A(1)pqr Fef . (34)

At second order equation (29) becomes

T (0)(αβ|λ F (2)|γ )λ − ∇(α| F (2)|βγ ) − iσ g

(αβ| F (0)|γ )g + iγ

6σ pqref

(αβ| A(1)pqr H(0)g

ef[−iσg|γ )φλφ

] = 0 (35)

or, more clearly

196 S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198

T (0)(αβ|λ

[− iγ

12σ pqref |γ )λ Fef A(1)

pqr

]+ iγ

12σ pqref

(αβ| Fef ∇|γ ) A(1)pqr − iγ

12σ pqref

(αβ| A(1)pqr∇|γ ) Fef

− iσ g(αβ| F (0)|γ )g + i

γ

6σ pqref

(αβ| A(1)pqr H(0)g

ef[−iσg|γ )φλφ

] = 0. (36)

Using (21) generates two solvable terms and we also set up a cancellation. We also must use

σ pqref(αβ|σe|γ )φ = −σ pqref

φ(α|σe|βγ ). (37)

Hence, we obtain

−iσ g(αβ| F (2)|γ )g + 2iγ σ g

(αβ|Ω(1)|γ )gef F (0)ef − γ

6σ g

(αβ|σ pqrg f |γ )φ A(1)pqr∇ f λ

φ + iγ

12σ pqref

(αβ| A(1)pqr

[−2iσ g |γ )φ H(0)gef

]

− iγ

6σ pqref

(αβ| A(1)pqr H(0)g

ef[−iσg|γ )φλφ

] = 0. (38)

The last two terms conveniently cancel. We find

iσ g(αβ| F (2)|γ )g = +2iγ σ

g(αβ|Ω

(1)|γ )gef F (0)ef − γ

6σ g

(αβ|σ pqrg f |γ )φ A(1)pqr∇ f λ

φ (39)

or

F (2)γ g = +2γΩ(1)

γ gef F (0)ef + iγ

6σ pqr

gfγ φ A(1)

pqr∇ f λφ. (40)

Finally for completeness, we consider the derivatives, ∇α F (2)bc and ∇(α| F (2)|β)g . We have

∇(α| F |β)d − Tαβg F gd − Tαβ

λ Fλd + T(α|d g F g|β) + T(α|dλ Fλ|β) = 0 (41)

and

∇α Fbc = ∇[b| Fα|c] + Tα[b| g F g|c] + Tα[b|λ Fλ|c] − Tbcg F gα + Tbc

λ Fλα. (42)

Substitution of (31), (32), (33), (34) and (40), as well as other results quoted in [2] and

T (2)αβ

λ = − iγ

12σ pqref

αβ A(1)pqr T (0)λ

ef (43)

give, respectively, for (41) and (42),

−iσd(α|φ∇|β)λφ |(2) + 2γ∇(α|Ω(1)|β)def F ef + 4iγΩ(1)

(α|def σe|β)φ∇ f λφ − 4iγΩ(1)

(α|def σg|β)φ H(0)gef

+ iγ

6σ pqr

df(α|φ

(∇|β) A(1)pqr

)[∇ f λφ] + iγ

6σ pqr

df(α|φ A(1)

pqr[∇|β)∇ f λ

φ] − iσ g

αβ F (2)gd + γ

12σ pqref

αβσdλφ A(1)pqr Tef

λλφ

− 2γ T (0)αβ

λΩ(1)λdef F (0)ef − i

γ

6T (0)

αβλσ pqr

dfλ f A(1)

pqr∇ f λφ − iσg(α|φλφ T (2)|β)d

g = 0, (44)

∇α Fbc = 2γ∇[b|Ω(1)α|c]ef F ef − i

γ

6σ pqr [b| f

αφ A(1)pqr∇|c]∇ f λ

φ − iγ

6σ pqr [b| f

αφ

[∇|c] A(1)pqr

]∇ f λφ

+ T (2)α[b|g F (0)

g|c] − iT (2)α[b|λσ|b]λφλφ + 2γ T (0)g

bcΩαgef F ef + iγ

6σ pqr

gfλφ A(1)

pqr T (2)bc

g∇ f λf − iγgαφλφ T (2)

bcg

− iγ

12σ pqref

λα A(1)pqr F (0)

ef T (0)bc

λ. (45)

4. Modified commutator

Finally, we note how the commutator (4) is modified. We have

[∇α,∇β} = − iγ

12σ pqref

αβ A(1)pqr

[Tef

γ ∇γ + 2H(0)ef

g∇g + 1

2R(0)

efmn Mmn + i F (0)

efI t I

]. (46)

But

2H(0)ef

g = −T (0)ef

g, (47)

and

R(0)ef

mn Mmn = −1

8R(0)

ef λδσmn

δλ (48)

so we get the modified geometry proportional to σ 5 as follows:

[∇α,∇β}|(2) = − iγ

12σ pqref

αβ A(1)pqr

[T (0)

efγ ∇γ − T (0)

efg∇g − 1

16R(0)

ef λδσmn

δλMmn − i F (0)

efI t I

]. (49)

A point conceptually important can be stressed here by observing that the latter expression agrees in its structure with the conjecturein Ref. [5] for supersymmetric Yang–Mills couplings.

S. Bellucci, D. O’Reilly / Physics Letters B 666 (2008) 193–198 197

5. Conclusion

The geometrical methods currently known as deformations [3] and the constraints often referred to as beta function favored con-straints [10–12] allowed for the determination of the most general higher derivative Yang–Mills action to the third order, which is globallysupersymmetric and Lorentz covariant in D = 10 spacetime (see e.g. [13]), a result which is important for topologically nontrivial gaugeconfigurations of the vector field, such as, for instance, in the case of compactified string theories on manifolds with topologically nontriv-ial properties. Building upon our previous works on the subject, we have been able here to provide a consistent inclusion of Yang–Millsmatter fields into string corrected (deformed) D = 10, N = 1 supergravity. Our solution to the Bianchi identities, obtained within theframework of the modified beta function favored constraints, holds to the second order in the string slope parameter γ and includesalso the Yang–Mills fields. We obtained as well a consistent solution in the torsion, curvature and H sectors with a Yang–Mills modifiedsupercurrent Aabc . Following a technique we developed in earlier papers, we also found a solution in the F sector and gave an explicitformula for the modification induced in the commutator expression.

Acknowledgement

This work of has been supported in part by the European Community Human Potential Program under contract MRTN-CT-2004-005104“Constituents, fundamental forces and symmetries of the universe”.

Appendix A

For the sake of extra caution as the Bianchi identity for example for Hαbd is long, we can write out the full version as follows:

1

2∇(α| H |βγ )d

(γ γ ;ββ;γ β) − ∇d HαβγOrder(2) − 1

2T (0)

(αβ|λH(γ γ )λ|γ )d − 1

2T (0)

(αβ|λH(γ β)λ|γ )d − 1

2T (0)

(αβ|λH(ββ)λ|γ )d

− 1

2T (γ )

(αβ|λH(β)λ|γ )d − 1

2T (γ )

(αβ|λH(γ )λ|γ )d − 1

2T (β)

(αβ|λH(β)λ|γ )d − 1

2T (β)

(αβ|λH(γ )λ|γ )d − 1

2T (γ γ )

(αβ|λH(0)λ|γ )d

− 1

2T (γ β)

(αβ|λH(0)λ|γ )d − 1

2T (ββ)

(αβ|λH(0)λ|γ )d − 1

2T (0)

(αβ|g H(γ γ )g|γ )d − 1

2T (0)

(αβ|g H(γ β)g|γ )d − 1

2T (0)

(αβ| g H(ββ)g|γ )d

− 1

2T (γ )

(αβ|g H(β)g|γ )d − 1

2T (β)

(αβ| g H(γ )g|γ )d + 1

2T (γ γ )

d(α|g H(0)g|βγ ) + 1

2T (γ β)

d(α| g H(0)g|βγ ) + 1

2T (ββ)

d(α|g H(0)g|βγ )

+ 1

2T (γ )

d(α|g H(γ )g|βγ ) + 1

2T (γ )

d(α| g H(β)g|βγ ) + 1

2T (β)

d(α|g H(γ )g|βγ ) + 1

2T (β)

d(α| g H(β)g|βγ ) + 1

2T (0)

d(α| g H(γ γ )g|βγ )

+ 1

2T (0)

d(α|g H(γ β)g|βγ ) + 1

2T (0)

d(α|g H(ββ)g|βγ )

+ γ[

R(γ )(αβ|ef R(0)|γ )d

ef + R(β)(αβ|ef R(0)|γ )d

ef + R(0)(αβ|ef R(β)|γ )d

ef + R(0)(αβ|ef R(γ )|γ )d

ef ]+ β

[F (γ )

(αβ| F (0)|γ )d + F (β)(αβ| F (0)|γ )d + F (0)

(αβ| F (β)|γ )d + F (0)(αβ| F (γ )|γ )d

] = 0. (A.1)

Applying the first order constraints and allowing H ABC terms to drop out leaves the H̄ ABC contributions.

1

2∇(α| H |βγ )d

(ββ;γ β) − 1

2T (0)

(αβ|λH(γ β)λ|γ )d − 1

2T (0)

(αβ|λH(ββ)λ|γ )d − 1

2T (γ β)

(αβ|λH(0)λ|γ )d − 1

2T (γ β)

(αβ|λH(0)λ|γ )d

− 1

2T (0)

(αβ|g H(γ β)g|γ )d − 1

2T (0)

(αβ|g H(ββ)g|γ )d + 1

2T (γ β)

d(α|g H(0)g|βγ ) + 1

2T (ββ)

d(α|g H(0)g|βγ )

+ 1

2T (γ γ )

d(α|g H(0)g|βγ ) + γ

[R(β)

(αβ|ef R(0)|γ )def + R(0)

(αβ|ef R(β)|γ )def ] = 0. (A.2)

We therefore find as before that no order β2 terms exist. Also there is no need to make modifications other than the adjustment ofAabc in the H sector.

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